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❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

▲❡✈ ❇✉❦♦✈s❦ý

❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻

❏✉❧② ✶✶✱ ✷✵✶✻

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ♣♦ss❡ss❡s t❤❡ ♣r♦♣❡rt② ✭αi✮✱ i = 1, 2✱ s❡❡ ❬❆r❤❪✱ ✐❢ ❢♦r ❛♥② x ∈ X ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ {{xn,m}∞

m=0}∞ n=0

♦❢ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ x✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ {ym}∞

m=0

s✉❝❤ t❤❛t limm→∞ ym = x ❛♥❞ ✭α1✮ {xn,m : m ∈ ω} ⊆∗ {ym : m ∈ ω} ❢♦r ❡❛❝❤ n✱ ✭α2✮ {xn,m : m ∈ ω} ∩ {ym : m ∈ ω} ✐s ✐♥✜♥✐t❡ ❢♦r ❡❛❝❤ n✳ ■t ✐s ❦♥♦✇ t❤❛t ❢♦r ❈p(X) t❤❡ ♣r♦♣❡rt✐❡s ✭α2✮✱ ✭α3✮ ❛♥❞ ✭α4✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ s❡❡ ❬❙❝✸❪✳ ❚❤❡ s❡q✉❡♥❝❡ s❡❧❡❝t✐♦♥ ♣r♦♣❡rt② ❙❙P✱ s❡❡ ❬❙❝✷❪✱ s❛②s✿ ❢♦r ❛♥② ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ ♦❢ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ ✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t ✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ♣♦ss❡ss❡s t❤❡ ♣r♦♣❡rt② ✭αi✮✱ i = 1, 2✱ s❡❡ ❬❆r❤❪✱ ✐❢ ❢♦r ❛♥② x ∈ X ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ {{xn,m}∞

m=0}∞ n=0

♦❢ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ x✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ {ym}∞

m=0

s✉❝❤ t❤❛t limm→∞ ym = x ❛♥❞ ✭α1✮ {xn,m : m ∈ ω} ⊆∗ {ym : m ∈ ω} ❢♦r ❡❛❝❤ n✱ ✭α2✮ {xn,m : m ∈ ω} ∩ {ym : m ∈ ω} ✐s ✐♥✜♥✐t❡ ❢♦r ❡❛❝❤ n✳ ■t ✐s ❦♥♦✇ t❤❛t ❢♦r ❈p(X) t❤❡ ♣r♦♣❡rt✐❡s ✭α2✮✱ ✭α3✮ ❛♥❞ ✭α4✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ s❡❡ ❬❙❝✸❪✳ ❚❤❡ s❡q✉❡♥❝❡ s❡❧❡❝t✐♦♥ ♣r♦♣❡rt② ❙❙P✱ s❡❡ ❬❙❝✷❪✱ s❛②s✿ ❢♦r ❛♥② x ∈ X ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ {{xn,m}∞

m=0}∞ n=0 ♦❢ s❡q✉❡♥❝❡s

❝♦♥✈❡r❣✐♥❣ t♦ x✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ {mn}∞

n=0 s✉❝❤ t❤❛t

limn→∞ xn,mn = x✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ s❡q✉❡♥❝❡ {fn}∞

n=0 ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ❛ s❡t X

q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ ❛ ❢✉♥❝t✐♦♥ f✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ {εn}∞

n=0 ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧s ❝♦♥✈❡r❣✐♥❣ t♦ 0 s✉❝❤ t❤❛t

(∀x ∈ X)(∃n0)(∀n ≥ n0) |fn(x) − f(x)| ≤ εn. ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ {fn}∞

n=0 ♦❢

❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❛❧s♦ q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ 0✱ ❬❇❘❘❪✳ ❆ t♦♣♦❧♦❣♦❝❛❧ s♣❛❝❡ ✐s ❛ ✲s♣❛❝❡ ✐❢ ❋ ✳ ❚❤❡♦r❡♠ ✶ ✭❘❡❝➟❛✇ ❬❘❪✮ ❆♥② ♣❡r❢❡❝t❧② ♥♦r♠❛❧ ◗◆✲s♣❛❝❡ ✐s ❛ ✲s♣❛❝❡✳ ❈♦r♦❧❧❛r② ✷ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❊✈❡r② s✉❜s❡t ✭✇✐t❤ t❤❡ s✉❜s❡t t♦♣♦❧♦❣②✮ ♦❢ ❛ ◗◆✲s♣❛❝❡ ✐s ❛ ◗◆✲s♣❛❝❡✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ s❡q✉❡♥❝❡ {fn}∞

n=0 ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ❛ s❡t X

q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ ❛ ❢✉♥❝t✐♦♥ f✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ {εn}∞

n=0 ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧s ❝♦♥✈❡r❣✐♥❣ t♦ 0 s✉❝❤ t❤❛t

(∀x ∈ X)(∃n0)(∀n ≥ n0) |fn(x) − f(x)| ≤ εn. ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ {fn}∞

n=0 ♦❢

❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❛❧s♦ q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ 0✱ ❬❇❘❘❪✳ ❆ t♦♣♦❧♦❣♦❝❛❧ s♣❛❝❡ X ✐s ❛ σ✲s♣❛❝❡ ✐❢ ❋σ(X) = Gδ(X)✳ ❚❤❡♦r❡♠ ✶ ✭❘❡❝➟❛✇ ❬❘❪✮ ❆♥② ♣❡r❢❡❝t❧② ♥♦r♠❛❧ ◗◆✲s♣❛❝❡ ✐s ❛ σ✲s♣❛❝❡✳ ❈♦r♦❧❧❛r② ✷ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❊✈❡r② s✉❜s❡t ✭✇✐t❤ t❤❡ s✉❜s❡t t♦♣♦❧♦❣②✮ ♦❢ ❛ ◗◆✲s♣❛❝❡ ✐s ❛ ◗◆✲s♣❛❝❡✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ✇◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ {fn}∞

n=0

♦❢ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❤❛s ❛ s✉❜s❡q✉❡♥❝❡ {fnk}∞

k=0 q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣✐♥❣ t♦ 0✱ ❬❇❘❘❪✳

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r ✐❢ ❡✈❡r② ♣❡r❢❡❝t s✉❜s❡t ♦❢ ✐s ♠❡❛❣❡r✳ ❚❤❡♦r❡♠ ✸ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❆♥② ✇◗◆✲s♣❛❝❡ ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭ ✮✱ ✱ ❙❙P✱ ✇◗◆✱ ◗◆✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ♦r ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❜② ❧♦✇❡r ♦r ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ♦♥❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❡❡ ❬❇❪✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭ ✮✱ ✱ ❙❙P✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❛❧❧ ❝❛s❡s ♦❢ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ❜② q✉❛s✐✲♥♦r♠❛❧ ❝♦♥✈❡r❣❡♥❝❡✳ ❈♦♠♣❛r❡ ❬❇❙❪✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ✇◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ {fn}∞

n=0

♦❢ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❤❛s ❛ s✉❜s❡q✉❡♥❝❡ {fnk}∞

k=0 q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣✐♥❣ t♦ 0✱ ❬❇❘❘❪✳

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r ✐❢ ❡✈❡r② ♣❡r❢❡❝t s✉❜s❡t ♦❢ X ✐s ♠❡❛❣❡r✳ ❚❤❡♦r❡♠ ✸ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❆♥② ✇◗◆✲s♣❛❝❡ ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭ ✮✱ ✱ ❙❙P✱ ✇◗◆✱ ◗◆✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ♦r ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❜② ❧♦✇❡r ♦r ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ♦♥❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❡❡ ❬❇❪✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭ ✮✱ ✱ ❙❙P✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❛❧❧ ❝❛s❡s ♦❢ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ❜② q✉❛s✐✲♥♦r♠❛❧ ❝♦♥✈❡r❣❡♥❝❡✳ ❈♦♠♣❛r❡ ❬❇❙❪✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ✇◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ {fn}∞

n=0

♦❢ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❤❛s ❛ s✉❜s❡q✉❡♥❝❡ {fnk}∞

k=0 q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣✐♥❣ t♦ 0✱ ❬❇❘❘❪✳

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r ✐❢ ❡✈❡r② ♣❡r❢❡❝t s✉❜s❡t ♦❢ X ✐s ♠❡❛❣❡r✳ ❚❤❡♦r❡♠ ✸ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❆♥② ✇◗◆✲s♣❛❝❡ ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭αi✮✱ i = 1, 2✱ ❙❙P✱ ✇◗◆✱ ◗◆✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ∗ ♦r ∗ ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❜② ❧♦✇❡r ♦r ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ♦♥❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❡❡ ❬❇❪✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭αi✮✱ i = 1, 2✱ ❙❙P✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ QN− ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❛❧❧ ❝❛s❡s ♦❢ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ❜② q✉❛s✐✲♥♦r♠❛❧ ❝♦♥✈❡r❣❡♥❝❡✳ ❈♦♠♣❛r❡ ❬❇❙❪✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❡♦r❡♠ ✹ ✭❙❝❤❡❡♣❡rs ❬❙❝✷❪✱ ❇✉❦♦✈s❦ý ✕ ❍❛❧❡➨ ❬❇❍❪✱ ❇✉❦♦✈s❦ý ✕ ➆✉♣✐♥❛ ❬❇❙❪✱ ❙❛❦❛✐ ❬❙❛✷❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ ❈p(X) ♣♦ss❡ss❡s t❤❡ ✭α1✮ ♣r♦♣❡rt②✱ ✷✮ ❈p(X) ♣♦ss❡ss❡s t❤❡ ◗◆✲❙❙P✱ ✸✮ ❈p(X) ♣♦ss❡ss❡s ◗◆✲✭α2✮ ♣r♦♣❡rt②✱ ✹✮ ❈p(X) ♣♦ss❡ss❡s ✭α2✮∗ ♣r♦♣❡rt②✱ ✺✮ ❈p(X) ♣♦ss❡ss❡s t❤❡ ❙❙P∗✱ ✻✮ X ✐s ❛ ◗◆✲s♣❛❝❡✱ ❈♦r♦❧❧❛r② ✺ ▲❡t ❜❡ ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦✐❝❛❧ s♣❛❝❡ s✉❝❤ t❤❛t ❈ ♣♦ss❡ss❡s t❤❡ ✭ ✮ ♣r♦♣❡rt②✳ ❚❤❡♥ ✶✮ ✐s ❛ ♣❡r❢❡❝t❧② ♠❡❛❣❡r ✲s♣❛❝❡✱ ✷✮ ❢♦r ❡✈❡r② s✉❜s❡t ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ s✉❜❡t t♦♣♦❧♦❣② t❤❡ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❈ ♣♦ss❡ss❡s t❤❡ ✭ ✮ ♣r♦♣❡rt②✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❡♦r❡♠ ✹ ✭❙❝❤❡❡♣❡rs ❬❙❝✷❪✱ ❇✉❦♦✈s❦ý ✕ ❍❛❧❡➨ ❬❇❍❪✱ ❇✉❦♦✈s❦ý ✕ ➆✉♣✐♥❛ ❬❇❙❪✱ ❙❛❦❛✐ ❬❙❛✷❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ ❈p(X) ♣♦ss❡ss❡s t❤❡ ✭α1✮ ♣r♦♣❡rt②✱ ✷✮ ❈p(X) ♣♦ss❡ss❡s t❤❡ ◗◆✲❙❙P✱ ✸✮ ❈p(X) ♣♦ss❡ss❡s ◗◆✲✭α2✮ ♣r♦♣❡rt②✱ ✹✮ ❈p(X) ♣♦ss❡ss❡s ✭α2✮∗ ♣r♦♣❡rt②✱ ✺✮ ❈p(X) ♣♦ss❡ss❡s t❤❡ ❙❙P∗✱ ✻✮ X ✐s ❛ ◗◆✲s♣❛❝❡✱ ❈♦r♦❧❧❛r② ✺ ▲❡t X ❜❡ ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦✐❝❛❧ s♣❛❝❡ s✉❝❤ t❤❛t ❈p(X) ♣♦ss❡ss❡s t❤❡ ✭α1✮ ♣r♦♣❡rt②✳ ❚❤❡♥ ✶✮ X ✐s ❛ ♣❡r❢❡❝t❧② ♠❡❛❣❡r σ✲s♣❛❝❡✱ ✷✮ ❢♦r ❡✈❡r② s✉❜s❡t Y ⊆ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ s✉❜❡t t♦♣♦❧♦❣② t❤❡ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❈p(Y ) ♣♦ss❡ss❡s t❤❡ ✭α1✮ ♣r♦♣❡rt②✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ ❝♦✉♥t❛❜❧❡ ❢❛♠✐❧② {Un : n ∈ ω} ♦❢ s✉❜s❡t ♦❢ ❛ s❡t X ✐s ❛ γ✲❝♦✈❡r ✐❢ Un = X ❢♦r ❡❛❝❤ n ❛♥❞ t❤❡ s❡t {n ∈ ω : x / ∈ Un} ✐s ✜♥✐t❡ ❢♦r ❡❛❝❤ x ∈ X✳ ❚❤❡♦r❡♠ ✻ ✭❇✉❦♦✈s❦ý ✕ ❍❛❧❡➨ ❬❇❍❪✱ ❙❛❦❛✐ ❬❙❛✶❪✮ ❆ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢❛♠✐❧② ♦❢ ♦♣❡♥ γ✲❝♦✈❡rs ♦❢ X ♣♦ss❡ss❡s t❤❡ ❝♦✈❡r✐♥❣ ✭α1✮ ♣r♦♣❡rt②✱ ✐✳❡✳✱ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ {Un}∞

n=0 ♦❢ ♦♣❡♥ γ✲❝♦✈❡rs t❤❡r❡

❡①✐st ✜♥✐t❡ s❡ts Vn ⊆ Un s✉❝❤ t❤❛t

n(Un \ Vn) ✐s ❛ γ✲❝♦✈❡r✳

❆ s❡t ✐s ❡✈❡♥t✉❛❧❧② ❜♦✉♥❞❡❞ ✐❢ ✐s ❜♦✉♥❞❡❞ ✐♥ t❤❡ ♣r❡♦r❞❡r ❚❤❡♦r❡♠ ✼ ✭❚s❛❜❛♥ ✕ ❩❞♦♠s❦②② ❬❚❩❪✮ ❆ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❇♦r❡❧ ♠❡❛s✉r❛❜❧❡ ✐♠❛❣❡ ♦❢ ✐♥t♦ t❤❡ ❇❛✐r❡ s♣❛❝❡ ✐s ❡✈❡♥t✉❛❧❧② ❜♦✉♥❞❡❞✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❆ ❝♦✉♥t❛❜❧❡ ❢❛♠✐❧② {Un : n ∈ ω} ♦❢ s✉❜s❡t ♦❢ ❛ s❡t X ✐s ❛ γ✲❝♦✈❡r ✐❢ Un = X ❢♦r ❡❛❝❤ n ❛♥❞ t❤❡ s❡t {n ∈ ω : x / ∈ Un} ✐s ✜♥✐t❡ ❢♦r ❡❛❝❤ x ∈ X✳ ❚❤❡♦r❡♠ ✻ ✭❇✉❦♦✈s❦ý ✕ ❍❛❧❡➨ ❬❇❍❪✱ ❙❛❦❛✐ ❬❙❛✶❪✮ ❆ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢❛♠✐❧② ♦❢ ♦♣❡♥ γ✲❝♦✈❡rs ♦❢ X ♣♦ss❡ss❡s t❤❡ ❝♦✈❡r✐♥❣ ✭α1✮ ♣r♦♣❡rt②✱ ✐✳❡✳✱ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ {Un}∞

n=0 ♦❢ ♦♣❡♥ γ✲❝♦✈❡rs t❤❡r❡

❡①✐st ✜♥✐t❡ s❡ts Vn ⊆ Un s✉❝❤ t❤❛t

n(Un \ Vn) ✐s ❛ γ✲❝♦✈❡r✳

❆ s❡t A ⊆ ωω ✐s ❡✈❡♥t✉❛❧❧② ❜♦✉♥❞❡❞ ✐❢ A ✐s ❜♦✉♥❞❡❞ ✐♥ t❤❡ ♣r❡♦r❞❡r f ≤∗ g ≡ (∃n0)(∀n ≥ n0) f(n) ≤ g(n). ❚❤❡♦r❡♠ ✼ ✭❚s❛❜❛♥ ✕ ❩❞♦♠s❦②② ❬❚❩❪✮ ❆ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❇♦r❡❧ ♠❡❛s✉r❛❜❧❡ ✐♠❛❣❡ ♦❢ X ✐♥t♦ t❤❡ ❇❛✐r❡ s♣❛❝❡ ωω ✐s ❡✈❡♥t✉❛❧❧② ❜♦✉♥❞❡❞✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❡♦r❡♠ ✽ ✭❋r❡♠❧✐♥ ❬❋r❪✱ ❙❝❤❡❡♣❡rs ❬❙❝✹❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ Cp(X) ♣♦ss❡ss❡s t❤❡ ✭α2✮ ♣r♦♣❡rt②✱ ✷✮ Cp(X) ♣♦ss❡ss❡s t❤❡ ❙❙P✱ ✸✮ X ✐s ❛ ✇◗◆✲s♣❛❝❡✳ ❆ ✲❝♦✈❡r ✐s s❤r✐♥❦❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ✲❝♦✈❡r t❤❛t ✐s ❛ r❡✜♥❡♠❡♥t ♦❢ ✳ ✐s t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ♦♣❡♥ s❤r✐♥❦❛❜❧❡ ✲❝♦✈❡rs✳ ❚❤❡♦r❡♠ ✾ ✭❇✉❦♦✈s❦ý ✕ ❍❛❧❡➨ ❬❇❍❪✮ ■❢ ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ ✐s ❛ ✇◗◆✲s♣❛❝❡✱ ✷✮ ✐s ❛♥ ✲s♣❛❝❡✳ ❈♦r♦❧❧❛r② ✶✵ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✱ ❈ ♣♦ss❡ss❡s t❤❡ ✭ ✮ ♣r♦♣❡rt② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛♥ ✲s♣❛❝❡✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❡♦r❡♠ ✽ ✭❋r❡♠❧✐♥ ❬❋r❪✱ ❙❝❤❡❡♣❡rs ❬❙❝✹❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ Cp(X) ♣♦ss❡ss❡s t❤❡ ✭α2✮ ♣r♦♣❡rt②✱ ✷✮ Cp(X) ♣♦ss❡ss❡s t❤❡ ❙❙P✱ ✸✮ X ✐s ❛ ✇◗◆✲s♣❛❝❡✳ ❆ γ✲❝♦✈❡r U ✐s s❤r✐♥❦❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ γ✲❝♦✈❡r t❤❛t ✐s ❛ r❡✜♥❡♠❡♥t ♦❢ U✳ Γsh ✐s t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ♦♣❡♥ s❤r✐♥❦❛❜❧❡ γ✲❝♦✈❡rs✳ ❚❤❡♦r❡♠ ✾ ✭❇✉❦♦✈s❦ý ✕ ❍❛❧❡➨ ❬❇❍❪✮ ■❢ X ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ X ✐s ❛ ✇◗◆✲s♣❛❝❡✱ ✷✮ X ✐s ❛♥ S1(Γsh, Γ)✲s♣❛❝❡✳ ❈♦r♦❧❧❛r② ✶✵ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X✱ ❈p(X) ♣♦ss❡ss❡s t❤❡ ✭α2✮ ♣r♦♣❡rt② ✐❢ ❛♥❞ ♦♥❧② ✐❢ X ✐s ❛♥ S1(Γsh, Γ)✲s♣❛❝❡✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❡♦r❡♠ ✶✶ ✭❇✉❦♦✈s❦ý ❬❇❪✱ ❙❛❦❛✐ ❬❙❛✷❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ X ♣♦ss❡ss❡s t❤❡ ❙❙P∗✱ ✷✮ X ✐s ❛ ✇◗◆∗✲s♣❛❝❡✳ ❚❤❡♦r❡♠ ✶✷ ✭❇✉❦♦✈s❦② ❬❇❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ ✐s ❛♥ ✲s♣❛❝❡✱ ✷✮ ♣♦ss❡ss❡s t❤❡ ❙❙P ✱ ❈♦r♦❧❧❛r② ✶✸ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ ✐s ❛♥ ✲s♣❛❝❡✱ ✷✮ ✐s ❛ ✇◗◆ ✲s♣❛❝❡✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❡♦r❡♠ ✶✶ ✭❇✉❦♦✈s❦ý ❬❇❪✱ ❙❛❦❛✐ ❬❙❛✷❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ X ♣♦ss❡ss❡s t❤❡ ❙❙P∗✱ ✷✮ X ✐s ❛ ✇◗◆∗✲s♣❛❝❡✳ ❚❤❡♦r❡♠ ✶✷ ✭❇✉❦♦✈s❦② ❬❇❪✮ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ X ✐s ❛♥ S1(Γ, Γ)✲s♣❛❝❡✱ ✷✮ X ♣♦ss❡ss❡s t❤❡ ❙❙P∗✱ ❈♦r♦❧❧❛r② ✶✸ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✶✮ X ✐s ❛♥ S1(Γ, Γ)✲s♣❛❝❡✱ ✷✮ X ✐s ❛ ✇◗◆∗✲s♣❛❝❡✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❲❡ ❦♥♦✇ ❬❇❍❪✱ ❬❙❛✶❪ t❤❛t QN → S1(Γ, Γ) → wQN. ❇② ❆✳ ❉♦✇ ❬❉❪ ✐♥ ▲❛✈❡r ♠♦❞❡❧ ❢♦r ❇♦r❡❧ ❝♦♥❥❡❝t✉r❡ ✭ ✮ ✭ ✮ ❚❤✉s✱ ❜② ❚❤❡♦r❡♠s ✹ ❛♥❞ ✽ ✐♥ ▲❛✈❡r ♠♦❞❡❧ ❇② ❲✳ ❏✉st✱ ❆✳❲✳ ▼✐❧❧❡r✱ ▼✳ ❙❝❤❡❡♣❡rs ❛♥❞ P✳❏✳ ❙③❡♣t②❝❦✐ ❬❏▼❙❙❪ ❘❡❝➟❛✇ ❬❘❪ ♣r♦✈❡❞ t❤❛t ❇② ▼✐❧❧❡r ❬▼❪ ❍❡♥❝❡ ❛❣❛✐♥✱ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❲❡ ❦♥♦✇ ❬❇❍❪✱ ❬❙❛✶❪ t❤❛t QN → S1(Γ, Γ) → wQN. ❇② ❆✳ ❉♦✇ ❬❉❪ ✐♥ ▲❛✈❡r ♠♦❞❡❧ ❢♦r ❇♦r❡❧ ❝♦♥❥❡❝t✉r❡ ✭α2✮ → ✭α1✮. ❚❤✉s✱ ❜② ❚❤❡♦r❡♠s ✹ ❛♥❞ ✽ ✐♥ ▲❛✈❡r ♠♦❞❡❧ wQN ≡ S1(Γ, Γ) ≡ QN. ❇② ❲✳ ❏✉st✱ ❆✳❲✳ ▼✐❧❧❡r✱ ▼✳ ❙❝❤❡❡♣❡rs ❛♥❞ P✳❏✳ ❙③❡♣t②❝❦✐ ❬❏▼❙❙❪ ❘❡❝➟❛✇ ❬❘❪ ♣r♦✈❡❞ t❤❛t ❇② ▼✐❧❧❡r ❬▼❪ ❍❡♥❝❡ ❛❣❛✐♥✱ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❲❡ ❦♥♦✇ ❬❇❍❪✱ ❬❙❛✶❪ t❤❛t QN → S1(Γ, Γ) → wQN. ❇② ❆✳ ❉♦✇ ❬❉❪ ✐♥ ▲❛✈❡r ♠♦❞❡❧ ❢♦r ❇♦r❡❧ ❝♦♥❥❡❝t✉r❡ ✭α2✮ → ✭α1✮. ❚❤✉s✱ ❜② ❚❤❡♦r❡♠s ✹ ❛♥❞ ✽ ✐♥ ▲❛✈❡r ♠♦❞❡❧ wQN ≡ S1(Γ, Γ) ≡ QN. ❇② ❲✳ ❏✉st✱ ❆✳❲✳ ▼✐❧❧❡r✱ ▼✳ ❙❝❤❡❡♣❡rs ❛♥❞ P✳❏✳ ❙③❡♣t②❝❦✐ ❬❏▼❙❙❪ if t = b, then S1(Γ, Γ) QN. ❘❡❝➟❛✇ ❬❘❪ ♣r♦✈❡❞ t❤❛t ❇② ▼✐❧❧❡r ❬▼❪ ❍❡♥❝❡ ❛❣❛✐♥✱ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❲❡ ❦♥♦✇ ❬❇❍❪✱ ❬❙❛✶❪ t❤❛t QN → S1(Γ, Γ) → wQN. ❇② ❆✳ ❉♦✇ ❬❉❪ ✐♥ ▲❛✈❡r ♠♦❞❡❧ ❢♦r ❇♦r❡❧ ❝♦♥❥❡❝t✉r❡ ✭α2✮ → ✭α1✮. ❚❤✉s✱ ❜② ❚❤❡♦r❡♠s ✹ ❛♥❞ ✽ ✐♥ ▲❛✈❡r ♠♦❞❡❧ wQN ≡ S1(Γ, Γ) ≡ QN. ❇② ❲✳ ❏✉st✱ ❆✳❲✳ ▼✐❧❧❡r✱ ▼✳ ❙❝❤❡❡♣❡rs ❛♥❞ P✳❏✳ ❙③❡♣t②❝❦✐ ❬❏▼❙❙❪ if t = b, then S1(Γ, Γ) QN. ❘❡❝➟❛✇ ❬❘❪ ♣r♦✈❡❞ t❤❛t there exists an uncountable S1(Γ, Γ)−space. ❇② ▼✐❧❧❡r ❬▼❪ It is consistent with ZFC that every σ−set is countable. ❍❡♥❝❡ ❛❣❛✐♥✱ S1(Γ, Γ) QN ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

▼❛r✐♦♥ ❙❝❤❡❡♣❡rs ❬❙❝✹❪ r❛✐s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈♦♥❥❡❝t✉r❡ ✶✹ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X S1(Γ, Γ) ≡ wQN, or X is S1(Γ, Γ)−space ≡ ❈p(X) possesses ✭α2✮ property. or S1(Γsh, Γ) ≡ S1(Γ, Γ). ❇② ♣r❡s❡♥t❡❞ r❡s✉❧ts t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈✳ ❍♦✇❡✈❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s st✐❧❧ ♦♣❡♥✿ Pr♦❜❧❡♠ ✶✺ ■s ✐t ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈ t❤❛t ❄

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

▼❛r✐♦♥ ❙❝❤❡❡♣❡rs ❬❙❝✹❪ r❛✐s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈♦♥❥❡❝t✉r❡ ✶✹ ❋♦r ❛ ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X S1(Γ, Γ) ≡ wQN, or X is S1(Γ, Γ)−space ≡ ❈p(X) possesses ✭α2✮ property. or S1(Γsh, Γ) ≡ S1(Γ, Γ). ❇② ♣r❡s❡♥t❡❞ r❡s✉❧ts t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈✳ ❍♦✇❡✈❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s st✐❧❧ ♦♣❡♥✿ Pr♦❜❧❡♠ ✶✺ ■s ✐t ❝♦♥s✐st❡♥t ✇✐t❤ ❩❋❈ t❤❛t wQN S1(Γ, Γ)❄

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❬❆r❤❪ ❆r❦❤❛♥❣❡❧✬s❦✐✟ ✙ ❆✳❱✳✱ Spektr qastot topologiqeskogo

prostranstva i klassifikaci prostranstv, DAN SSSR✱

✷✵✻✿✷ ✭✶✾✼✷✮✱ ✷✻✺✕✷✻✽✳ ❊♥❣❧✐s❤ tr❛♥s❧❛t✐♦♥✿ ❚❤❡ ❢r❡q✉❡♥❝② s♣❡❝tr✉♠ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❛♥❞ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ s♣❛❝❡s✱ ❙♦✈✐❡t ▼❛t❤✳ ❉♦❦❧✳ ✶✸ ✭✶✾✼✷✮✱ ✶✶✽✺✕✶✶✽✾✳ ❬❇❪ ❇✉❦♦✈s❦ý ▲✳✱ ❖♥ ✇◗◆∗ ❛♥❞ ✇◗◆∗ s♣❛❝❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✶✺✻ ✭✷✵✵✽✮✱ ✷✹✕✷✼✳ ❬❇❍❪ ❇✉❦♦✈s❦ý ▲✳ ❛♥❞ ❍❛❧❡➨ ❏✳✱ ◗◆✲s♣❛❝❡s✱ ✇◗◆✲s♣❛❝❡s ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✶✺✹ ✭✷✵✵✼✮✱ ✽✹✽✕✽✺✽✳ ❬❇❘❘❪ ❇✉❦♦✈s❦ý ▲✳✱ ❘❡❝➟❛✇ ■✳ ❛♥❞ ❘❡♣✐❝❦ý ▼✳✱ ❙♣❛❝❡s ♥♦t ❞✐st✐♥❣✉✐s❤✐♥❣ ♣♦✐♥t✇✐s❡ ❛♥❞ q✉❛s✐✲♥♦r♠❛❧ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✹✶ ✭✶✾✾✶✮✱ ✷✺✕✹✵✳ ❬❇❙❪ ❇✉❦♦✈s❦ý ▲✳ ❛♥❞ ➆✉♣✐♥❛ ❏✳✱ ▼♦❞✐✜❝❛t✐♦♥s ♦❢ ❙❡q✉❡♥❝❡ ❙❡❧❡❝t✐♦♥ Pr✐♥❝✐♣❧❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✶✻✵ ✭✷✵✶✸✮✱ ✷✸✺✻✕✷✸✼✵✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❬❉❪ ❉♦✇ ❆✳✱ ❚✇♦ ❝❧❛ss❡s ♦❢ ❋ré❝❤❡t✲❯r②s♦❤♥ s♣❛❝❡s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✸✶ ✭✶✾✾✵✮✱ ✷✹✶✕✷✹✼✳ ❬❋r❪ ❋r❡♠❧✐♥ ❉✳❍✳✱ ❙❙P ❛♥❞ ✇◗◆✱ ♣r❡♣r✐♥t ✷✵✵✷✳ ❬❏▼❙❙❪ ❏✉st ❲✳✱ ▼✐❧❧❡r ❆✳❲✳✱ ❙❝❤❡❡♣❡rs ▼✳ ❛♥❞ ❙③❡♣t②❝❦✐ P✳❏✳✱ ❈♦♠❜✐♥❛t♦r✐❝s ♦❢ ♦♣❡♥ ❝♦✈❡rs ✭■■✮✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✼✸ ✭✶✾✾✻✮✱ ✷✹✶✕✷✻✻✳ ❬▼❪ ▼✐❧❧❡r ❆✳❲✳✱ ❙♣❡❝✐❛❧ s✉❜s❡ts ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✱ ✐♥✿ ❍❛♥❞❜♦♦❦ ♦❢ ❙❡t ❚❤❡♦r❡t✐❝ ❚♦♣♦❧♦❣② ✭❑✉♥❡♥ ❑✳ ❛♥❞ ❱❛✉❣❤❛♥ ❏✳❊✳✱ ❡❞s✳✮✱ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ❆♠st❡r❞❛♠✱ ✶✾✽✹✱ ✷✵✶ ✕ ✷✸✺✳ ❬❘❪ ❘❡❝➟❛✇ ■✳✱ ▼❡tr✐❝ s♣❛❝❡s ♥♦t ❞✐st✐♥❣✉✐s❤✐♥❣ ♣♦✐♥t✇✐s❡ ❛♥❞ q✉❛s✐♥♦r♠❛❧ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s✱ ❇✉❧❧✳ ❆❝❛❞✳ ❙❝✐✳ P♦❧♦♥❛✐s ✹✺ ✭✶✾✾✼✮✱ ✷✽✼ ✕ ✷✽✾✳ ❬❙❛✶❪ ❙❛❦❛✐ ▼✳✱ ❚❤❡ s❡q✉❡♥❝❡ s❡❧❡❝t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ ❈p(X)✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✶✺✹ ✭✷✵✵✼✮✱ ✺✺✷✕✺✻✵✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❬❙❛✷❪ ❙❛❦❛✐ ▼✳✱ ❙❡❧❡❝t✐♦♥ ♣r✐♥❝✐♣❧❡s ❛♥❞ ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✱✱ ❈♦❧❧♦q✉✐✉♠ ▼❛t❤❡♠❛t✐❝✉♠ ✶✶✼ ✭✷✵✵✾✮✱ ✷✺✶✕✷✺✻✳ ❬❙❝✶❪ ❙❝❤❡❡♣❡rs ▼✳✱ ❈♦♠❜✐♥❛t♦r✐❝s ♦❢ ♦♣❡♥ ❝♦✈❡rs ■✿ ❘❛♠s❡② t❤❡♦r②✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✻✾ ✭✶✾✾✻✮✱ ✸✶✕✻✷✳ ❬❙❝✷❪ ❙❝❤❡❡♣❡rs ▼✳✱ ❆ s❡q✉❡♥t✐❛❧ ♣r♦♣❡rt② ♦❢ Cp(X) ❛♥❞ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♦❢ ❍✉r❡✇✐❝③✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✷✺ ✭✶✾✾✼✮✱ ✷✼✽✾✕✷✼✾✺✳ ❬❙❝✸❪ ❙❝❤❡❡♣❡rs ▼✳✱ ❈p(X) ❛♥❞ ❆r❤❛♥❣❡❧✬s❦✐✟ ✙✬s αi✲s♣❛❝❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✽✾ ✭✶✾✾✽✮✱ ✷✻✺✕✷✼✺✳ ❬❙❝✹❪ ❙❝❤❡❡♣❡rs ▼✳✱ ❙❡q✉❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡s ✐♥ Cp(X) ❛♥❞ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt②✱ ❊❛st✲❲❡st ❏✳ ▼❛t❤✳ ✶ ✭✶✾✾✾✮✱ ✷✵✼✕✷✶✹✳ ❬❚❩❪ ❚s❛❜❛♥ ❇✳ ❛♥❞ ❩❞♦♠s❦②② ▲✳✱ ❍❡r❡❞✐t❛r② ❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ ❆r❦❤❛♥❣❡❧✬s❦✐✟ ✙ s❤❡❛❢ ❛♠❛❧❣❛♠❛t✐♦♥s✱ ❏✳ ❊✉r✳ ▼❛t❤✳ ❙♦❝✳ ✶✹ ✭✷✵✶✷✮✱ ✸✺✸✕✸✼✷✳

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

❚❤❛♥❦ ❨♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥

■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡s✱ ❯♥✐✈❡rs✐t② ♦❢ P✳❏✳ ➆❛❢ár✐❦ ❏❡s❡♥♥á ✺✱ ✵✹✵ ✵✶ ❑♦➨✐❝❡✱ ❙❧♦✈❛❦✐❛ ❡✲♠❛✐❧✿ ❧❡✈✳❜✉❦♦✈s❦②❅✉♣❥s✳s❦

▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ Cp(X) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X